Question

The number of ways in which an arrangement of 4 letters of the word PROPORTION can be made is

• A. $700$
• B. $750$
• C. $758$
• D. $800$

Answer 1

The correct option is C $758$

The letters are:
P R O T I N
P R O
O
Thus, number of permutations taken 4 at a time is the coefficient of $x^4$ in:
$4!\times (1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}).{(1+\frac{x}{1!}+\frac{x^2}{2!})}^2.{(1+\frac{x}{1!})}^3=>(6+6x+3x^2+x^3).{(2+2x+x^2)}^2.{(1+x)}^3=>(6+6x+3x^2+x^3).(4+8x^2+x^4+8x+4x^3).(1+3x+3x^2+x^3)=>\left\{\right(6+24+24+8)+(24+48+24+4).3+(48+48+12).3+(48+24\left)\right\}=(62+300+324+72)=758$
Hence, (c) is correct.